In a regular square pyramid , the base is a square of side . The apex is directly above the center of the base, and . Find the dihedral angle along edge between planes and .

The dihedral angle satisfies (so ).

Solid Geometry Solution – Problem 15: Find the dihedral angle along edge between planes and

Question

In a regular square pyramid $P-ABCD$ , the base $ABCD$ is a square of side $2$ . The apex $P$ is directly above the center $O$ of the base, and $PO=2$ . Find the dihedral angle along edge $AB$ between planes $PAB$ and $ABCD$ .

Step-by-step solution

Set up coordinates with the base in $z=0$ and center at the origin: $A(-1,-1,0),\ B(1,-1,0),\ C(1,1,0),\ D(-1,1,0),\ P(0,0,2).$ Plane $ABCD$ is $z=0$ , so a normal vector is $\vec n_1=(0,0,1)$ . For plane $PAB$ , use two direction vectors in the plane: $\overrightarrow{AB}=(2,0,0),\qquad \overrightarrow{AP}=(1,1,2).$ A normal vector is $\vec n_2=\overrightarrow{AB}\times\overrightarrow{AP}=(0,-4,2)\sim(0,-2,1).$ Let $\theta$ be the dihedral angle (the angle between the two planes). Then $\cos\theta=\frac{|\vec n_1\cdot\vec n_2|}{|\vec n_1||\vec n_2|}= rac{1}{\sqrt5}.$

Final answer

The dihedral angle $\theta$ satisfies $\cos\theta=\dfrac{1}{\sqrt5}$ (so $\theta=\arccos\dfrac{1}{\sqrt5}$ ).

Marking scheme

1. Checkpoints (max 7 pts total)

Coordinate setup (2 pts): Place the square base of side $2$ in $z=0$ with correct center and height $2$ .
Normal vectors (3 pts): Find $\vec n_1$ for the base plane and $\vec n_2=\overrightarrow{AB}\times\overrightarrow{AP}$ for plane $PAB$ .
Angle computation (2 pts): Compute $\cos\theta=\frac{|\vec n_1\cdot\vec n_2|}{|\vec n_1||\vec n_2|}=\frac{1}{\sqrt5}$ .

2. Zero-credit items

Using a 2D triangle angle in a face as the dihedral angle without justification.
Giving $\cos\theta$ without any normal-vector work.

3. Deductions

Cross-product mistake (-1): incorrect $\overrightarrow{AB}\times\overrightarrow{AP}$ .
Normalization/arithmetic error (-1): incorrect computation of $|\vec n_2|$ .

Solid Geometry – Problem 15: Find the dihedral angle along edge between planes and